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Search: id:A059933
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| A059933 |
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Goodstein sequence with a(2)=16: to calculate a(n+1), write a(n) in the hereditary representation base n, then bump the base to n+1, then subtract 1. |
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+0
8
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| 16, 7625597484986, 50973998591214355139406377, 19916489515870532960258562190639398471599239042185934648024761145811, 5103702287864892035208610181878203902270504134895451401860454182513968464023205038690962121196797
(list; graph; listen)
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OFFSET
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2,1
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COMMENT
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Goodstein's theorem shows that such a sequence is finite (i.e. it eventually stablizes and then decreases by 1 in each step until it reaches 0) for any starting point of a(2). In this case of a(2)=16, there seems little possibility of describing how incredibly large n must be for a(n)=0.
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REFERENCES
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Goodstein, R. L. "On the Restricted Ordinal Theorem." J. Symb. Logic 9, 33-41, 1944
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EXAMPLE
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a(2) = 16 = 2^(2^2) so a(3) = 3^(3^3)-1 = 7625597484986.
So a(3) = 2*3^(2*3^2 + 2*3 + 2) + 2*3^(2*3^2 + 2*3 + 1) + 2*3^(2*3^2 + 2*3) + 2*3^(2*3^2 + 1*3 + 2) + 2*3^(2*3^2 + 1*3 + 1) + 2*3^(2*3^2 + 1*3) + 2*3^(2*3^2 + 2) + 2*3^(2*3^2 + 1) + 2*3^(2*3^2) + 2*3^(3^2 + 2*3 + 2) + 2*3^(3^2 + 2*3 + 1) + 2*3^(3^2 + 2*3) + 2*3^(3^2 + 1*3 + 2) + 2*3^(3^2 + 1*3 + 1) + 2*3^(3^2 + 1*3) + 2*3^(3^2 + 2) + 2*3^(3^2 + 1) + 2*3^(3^2) + 2*3^(2*3 + 2) + 2*3^(2*3 + 1) + 2*3^(2*3) + 2*3^(1*3 + 2) + 2*3^(1*3 + 1) + 2*3^(1*3) + 2*3^(2) + 2*3^(1) + 2,
leading to a(4) = 2*4^(2*4^2 + 2*4 + 2) + 2*4^(2*4^2 + 2*4 + 1) + 2*4^(2*4^2 + 2*4) + 2*4^(2*4^2 + 1*4 + 2) + 2*4^(2*4^2 + 1*4 + 1) + 2*4^(2*4^2 + 1*4) + 2*4^(2*4^2 + 2) + 2*4^(2*4^2 + 1) + 2*4^(2*4^2) + 2*4^(4^2 + 2*4 + 2) + 2*4^(4^2 + 2*4 + 1) + 2*4^(4^2 + 2*4) + 2*4^(4^2 + 1*4 + 2) + 2*4^(4^2 + 1*4 + 1) + 2*4^(4^2 + 1*4) + 2*4^(4^2 + 2) + 2*4^(4^2 + 1) + 2*4^(4^2) + 2*4^(2*4 + 2) + 2*4^(2*4 + 1) + 2*4^(2*4) + 2*4^(1*4 + 2) + 2*4^(1*4 + 1) + 2*4^(1*4) + 2*4^(2) + 2*4^(1) + 1 = 2*(4^32 + 4^16 + 1)*(4^8 + 4^4 + 1)*(4^2 + 4*1)-1 = 50973998591214355139406377.
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CROSSREFS
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Cf. A056193, A056004, A057650, A056041.
Sequence in context: A116102 A013878 A058418 this_sequence A002488 A088469 A089170
Adjacent sequences: A059930 A059931 A059932 this_sequence A059934 A059935 A059936
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KEYWORD
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fini,nonn
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Feb 12 2001
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EXTENSIONS
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Definition corrected by njas, Mar 06 2006
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